Local restriction theorem and maximal Bochner-Riesz operators for the Dunkl transforms
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- by Feng Dai and Wenrui Ye PDF
- Trans. Amer. Math. Soc. 371 (2019), 641-679 Request permission
Abstract:
For the Dunkl transforms associated with the weight functions $h_\kappa ^2(x)=\prod _{j=1}^d |x_j|^{2\kappa _j}$, $\kappa _1,\cdots , \kappa _d\ge 0$ on $\mathbb {R}^d$, it is proved that if $p\ge 2+\frac 1{\lambda _\kappa }$ and $\lambda _\kappa :=\frac {d-1}2+\sum _{j=1}^d\kappa _j$, the maximal Bochner-Riesz operator $B_\ast ^\delta (h_\kappa ^2; f)$ order $\delta >0$ is bounded on the space $L^p(\mathbb {R}^d; h_\kappa ^2dx)$ if and only if $\delta >\delta _\kappa (p):=\max \{(2\lambda _\kappa +1) (\frac 12-\frac 1p)-\frac 12,0\}$. This extends a well known result of M. Christ for the classical Fourier transforms (Proc. Amer. Math. Soc. 95 (1985), 16–20). The proof relies on a new local restriction theorem for the Dunkl transforms, which is stronger than the corresponding global restriction theorem, but significantly more difficult to prove.References
Additional Information
- Feng Dai
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 660750
- Email: fdai@ualberta.ca
- Wenrui Ye
- Affiliation: School of Statistics, University of International Business and Economics, Chaoyang Qu, Beijing 100029, People’s Republic of China
- MR Author ID: 1036287
- Email: wye@uibe.edu.cn
- Received by editor(s): December 4, 2016
- Received by editor(s) in revised form: May 9, 2017
- Published electronically: June 26, 2018
- Additional Notes: This work was supported by NSERC Canada under grant RGPIN 04702 Dai. It was conducted when the second author was a PhD student at the University of Alberta.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 641-679
- MSC (2010): Primary 42B10, 42B15; Secondary 40A10, 33C10
- DOI: https://doi.org/10.1090/tran/7285
- MathSciNet review: 3885156